研究生一年级Algebra计划中

Commutative algebra

Commutative rings and their modules: prime and maximal ideals, localisation, Noetherian rings and Hilbert's basis theorem, integral extensions, the Nullstellensatz, primary decomposition and dimension.

math.algebra.commutative-algebra

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前置知识

Modules

29 项必需的前置知识（一直追溯到底）

你将学会

Localise rings and modules, and use localisation to reduce problems to local rings.
Prove Hilbert's basis theorem and Nakayama's lemma.
State and use the Nullstellensatz to relate ideals and algebraic sets.

定义

The spectrum Spec R is the set of prime ideals of R.
A ring is Noetherian if every ascending chain of ideals stabilises.
The localisation S⁻¹R formally inverts the elements of a multiplicative subset S.

定理

Hilbert's basis theorem: if R is Noetherian then so is R[x].
Nullstellensatz: over an algebraically closed field k, the maximal ideals of k[x₁, …, xₙ] correspond to the points of kⁿ.
Nakayama's lemma for finitely generated modules over local rings.
The Krull dimension of k[x₁, …, xₙ] is n.

常见误解

Thinking every ring is Noetherian; k[x₁, x₂, x₃, …] is not.
Confusing prime ideals with maximal ideals.

通向

Algebraic number theory Algebraic geometry

可选发现

Homological algebra Regular local rings are characterised homologically, as the Noetherian local rings of finite global dimension (Auslander–Buchsbaum–Serre).
← Rings, ideals and fields

学习路径

Graduate preparation